Describe an isomorphism between the equivalence class [f]∼ and the set ℝ.

Question

Let S be the set of differentiable functions ℝ→ℝ . Let ∼ be the relation defined on S by f1∼f2  if and only if  ∀x∈ℝ,f′1(x)=f′2(x). 

Let f∈S be a differentiable function. Describe an isomorphism between the equivalence class [f]∼ and the set ℝ.

Answer 1

May you clarify what f'1(x) and f'2(x) are? Are these the first and second derivatives?